On a general smoothly truncated regularization for variational piecewise constant image restoration: construction and convergent algorithms

Abstract

Image restoration is a typical inverse problem, and piecewise constant images have extensive applications in industry and business. Variational models with nonconvex, nonsmooth regularizations can achieve high-quality restorations with neat edges. In particular, a class of truncated potential functions effectively supports contrast-preserving restoration. However, these functions are not subdifferentially regular and thus yield no variational or convergence results for minimization algorithms. In this paper, we present a general smoothing scheme to overcome this nonregularity of the existing truncated regularizers. We also propose globally convergent algorithms to solve the noncoercive variational models with our new smoothly truncated regularizer (STR) functions by introducing a novel proximal term. The limit point of the iterative sequence is shown to be a -stationary point of the original objective function. We then give the implementation details for the inner subproblem by the alternating direction method of multipliers (ADMM). Numerical experiments are carried out to illustrate the good ability of the new regularizer to preserve neat edges and contrasts for piecewise constant images.